Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

learn more… | top users | synonyms

-1
votes
1answer
26 views

How to calculate frequency with clock signal is 500ps in digital logic?

How can i calculate frequency if clock signal 500ps. I know the only formula, that is T=1/f But i cant able to calculate, can ...
1
vote
2answers
136 views

Understanding inverse of a function

I was trying to understand the proof for the following proposition. Proposition: If $\{f_n\}$ is a sequence of $\bar{\mathbb{R}}$ valued measurable functions on $(X,\mathcal{M})$, then the functions ...
1
vote
1answer
67 views

find an inverse function of complicated one

Let $f:\mathbb{R}\rightarrow \mathbb{R}$: $$f(x) = \sin (\sin (x)) +2x$$ How to calculate the inverse of this function? So far i searched a lot in the internet but i didn't find any easy algorithm ...
0
votes
1answer
38 views

find the inverse of $\frac{1-e^t}{1+e^t}$

Hi I am trying to prove that the inverse of $f(t) = \frac{1-e^t}{1+e^t}$ is $F^{-1}(t) = \ln\left(\frac{1-t}{1+t} \right )$ But I don't quite know where to start? Do I just sub ...
1
vote
3answers
72 views

Proof Regarding Determinants of a Matrix

Prove the following statement: If $A$ is an $n$ by $n$ matrix, such that $\sum_{j = 1}^n a_{ij} = 0$, for all $1 ≤ i ≤ n$, then $\det A = 0$ too. (Sorry I don't know how to format this equation) ...
0
votes
2answers
38 views

Invertible matrix problem

Given three $n \times n$ matrices $A$, $B$ and $C$. Prove that if $AB+AC$ is an invertible matrix then $A$ is also an invertible matrix. How can this be possible? I found that $B=A^{-1}-C$ and when I ...
1
vote
1answer
37 views

Inverse Laplace tranform via the table formulas

In my inverse Laplace table there is this inversion "formula": $(1) \frac{1}{s-a} \rightarrow e^{at}$ I understand that $\mathcal{L}^{-1}[\frac{1}{s+4}] = \frac{1}{2}\sin(2t)$ But why can I not do ...
6
votes
1answer
53 views

Tan inverse summation

$$S=\sum\limits_{i=1}^{4}\tan^{-1} x_i$$ How to simplify this ? I think I will have to use this : but it looks too long a method . Is there a method or symmetrical way which yields ...
0
votes
0answers
30 views

Sherman–Morrison–Woodbury formula and hollow matrix

Suppose there are two matrices: $A_{n\times n}= \begin{bmatrix} a_0 & 0 &a_1 & \dots \\ 0 & a_1 & 0 &\dots \\ a_1 & 0 &a_2 & \dots \\ \vdots & \vdots & ...
3
votes
1answer
770 views

inverse of a covariance matrix 3x3

I have 2 pixels with size 1x3 called $A$ and $B$ and I have to compute the following equation: $$ A^T *(\Sigma+ I_3*\lambda)^{-1}*B $$ where $\Sigma$ is the covariance matrix (3x3) between vectors ...
3
votes
1answer
26 views

If $\sin^{-1}\frac{2a}{1+a^2}-\cos^{-1}\frac{1-b^2}{1+b^2}=\tan^{-1}\frac{2x}{1-x^2}$ then what is value of x?

If $\sin^{-1}\frac{2a}{1+a^2}-\cos^{-1}\frac{1-b^2}{1+b^2}=\tan^{-1}\frac{2x}{1-x^2}$ then what is value of x? Solution $\tan^{-1}x=\tan^{-1}a-\tan^{-1}b=\tan^{-1}\frac{a-b}{1+ab}$ ...
3
votes
1answer
42 views

Arithmetic modulo primes task

I'm dealing with a problem here. The problem is as follows: There is a set $Z_p=\{0,1,2,3,...,p-1\}$ where $p$ is a prime. From this set we form a new set $B=\{x+x^{-1}\mid x\in Z_p\}$, where the ...
2
votes
0answers
80 views

Closed form for elements of inverse matrix of lower triangular matrix of any size

If we have a lower triangular matrix $$A=\left(\begin{array}{rrrrr}a_{1,1}&0&0&\cdots&0\\a_{2,1}&a_{2,2}&0&\cdots&0\\a_{3,1} &1_{3,2}&a_{3,3}&\cdots&0\\ ...
0
votes
1answer
19 views

a matrix inverse problem

Given a matrix $X$, let $D$ be a diagonal matrix whose diagonal elements are row sums of $X$, let $I$ be an identity matrix. Now I have a resultant matrix of $Y=(I-X)^{-1}$, and I would like to ...
1
vote
1answer
23 views

Deriving an identity using the Woodbury matrix identity

I am working through an algorithm derivation in Kernel Adaptive Filtering: A Comprehensive Introduction by Liu, Principe and Haykin. The part I'm having trouble with is on page 104 if you have the ...
0
votes
1answer
12 views

Inversion of Boolean function Application

Asume you have a boolean function $f$ which takes $n$ parameters and gives $m$ results. In addition, you have a boolean function $g$ takes $p$ parameters and gives out $n$ results. You could ...
0
votes
0answers
48 views

Show that the inverse function to $f(x)=\int_{1}^{x}\frac{dt}{t}$ is differentiable

Show that the inverse function to $$f(x)=\int_{1}^{x}\frac{dt}{t}$$ is differentiable. I know that the integral is $\ln(x)$, but I don't know how to show that it is differentiable in a good way ...
4
votes
3answers
138 views

Calculate inverse of arbitrarily sized, lower triangle matrix with a specific pattern.

I have a matrix of the following form: $$A=\begin{bmatrix} 2 & 0 & 0 & 0 \\-1 & 2 & 0 & 0\\ 0 & -1 & 2 & 0\\ 0 & 0 & -1 & 2 \end{bmatrix}$$ which, in ...
1
vote
1answer
44 views

Cholesky, Inverse, and Determinant when updating the diagonal of a symmetric positive definite matrix

Suppose that $A$ is a symmetric positive definite matrix and assume its dimension $n$ is large. Let $I$ be the $n \times n$ identity matrix and $m \neq 0$ be a scalar. I'm interested in computing as ...
0
votes
1answer
52 views

Word problem about finding the inverse derivative

I have the following word problem. I need to find and interpret the meaning of the inverse derivative of a function. At a gas station, the function f(p) is the number of gallons of gasoline sold when ...
7
votes
3answers
686 views

How to derive compositions of trigonometric and inverse trigonometric functions?

To prove: $$\begin{align} \sin({\arccos{x}})&=\sqrt{1-x^2}\\ \cos{\arcsin{x}}&=\sqrt{1-x^2}\\ \sin{\arctan{x}}&=\frac{x}{\sqrt{1+x^2}}\\ \cos{\arctan{x}}&=\frac{1}{\sqrt{1+x^2}}\\ ...
0
votes
4answers
60 views

Calculate inverse of matrix

If $$A=\begin{bmatrix} -5 & 1 & 0 & 0\\ -19 & 4 & 0 & 0\\ 0 & 0 & 1 & 2\\ 0 & 0 & 3 & 5\\ \end{bmatrix}, $$ how do I calculate $A^{-1}$? Is there any ...
1
vote
1answer
25 views

Is it Possible to Develop an inverse function using the function it self

Is it Possible to Develop (taylor expansion) of an inverse function by knowing the function it self ? If Yes ,Can you illustrate with a simple function I know that we use the identity formula $$ ...
1
vote
2answers
44 views

Finding a Matrix from Determinants

I've stumbled upon this problem on my homework, and I have no clue how to do it, and haven't found any help online: If I'm understanding this correctly, then $det(M) = ad - cb + eh - gf$ ? What I ...
0
votes
1answer
22 views

Derivative of inverse function where inverse is known only numerically.

I have the following polynomial function in $\mathbb{R}$: $$f(x,a)=ax^3+x$$ However, $x$ also depends on $a$, so we should rather write: $$f(x,a)=ax(a)^3+x(a)$$ Now I need derivative of $f$ and ...
1
vote
2answers
43 views

Inverse of a special matrix

Is there easy (analytical) way to find the inverse of the following matrix, where $C$ is a vector? $$ \begin{bmatrix} 1 & C^\top \\ C & CC^\top \end{bmatrix} $$
2
votes
1answer
50 views

When is the inverse of a sparse matrix dense?

The question is basically stated in the title. Say $A$ is a sparse square matrix, then Is there any way to estimate the density of non-zero elements of $A^{-1}$? What properties of $A$ are ...
0
votes
1answer
34 views

Inverse of split functions

I don't know how to find the inverse is of a function when is split. Example, $\Bbb R_+$ is the set of positive real numbers. $f : \Bbb R \to \Bbb R_+$ $$f(x) = \begin{cases} 2-x & \text{if } ...
2
votes
2answers
11k views

Matrix is singular to working precision

I have a problem while evaluating inverse using inv in MATLAB. My matrix looks like this: ...
0
votes
0answers
24 views

check a matrix for positive semidefiniteness in the general case?

Here is the problem that I am facing. I know that for a positive semidefinite matrix to exist, the condition below must be satisfied: $x^\intercal Y x \geq 0$ where $x$ is a non-zero vector, i.e. a ...
1
vote
2answers
52 views

How to calculate the inverse of sum of a Kronecker product and a diagonal matrix

I want to calculate the inverse of a matrix of the form $S = (A\otimes B+C)$, where $A$ and $B$ are symetric and invertible, $C$ is a diagonal matrix with positive elements. Basically if the ...
0
votes
1answer
31 views

jacobian matrix of $f^{-1}$ at $(1,0,2)$ given $f(x,y,z)$

If $f(x,y,z)=(sin(xyz),(x+x^2)*cos(y),y)$ and $f$ has a local inverse in the neighborhood of $(0,1,1)$, how do I find the jacobian matrix of this inverse at $(1,0,2)$? I know from definition that ...
2
votes
3answers
54 views

Find the slope of the tangent line to the graph of $f^{-1}$

Given function $f$, find the slope of the line tangent to the graph $f^{-1}$ at the point on the graph $f^{-1}$. $f(x)=\sqrt{5x}$; $(4,\frac{16}{5})$? Here is what I have thus far: $f'(x)= ...
1
vote
3answers
80 views

How to solve an Inverse differentiation problem

If f is a one-to-one function where $f(3)=2$ and $f'(3)=6$, what is the value of $(f^{-1})'(2)$? I am not even sure where to start with this question. I was hoping someone can help $f$ of $3 =2$ and ...
1
vote
0answers
26 views

implementing modular multiplicative inverse.

I wish to implement the clifford cocks algorithm using GMP. In the encryption part: $c_1=t_1+at_1^{-1}\bmod n$. Following $( a b \bmod n ) = ((a \bmod n) \cdot (b \bmod n ))\bmod n$, I took the ...
0
votes
0answers
34 views

Integral of reciprocal of a piecewise linear function

Let, e.g. $$ f(x) = \begin{cases} x,\quad x<1, \\ 1,\quad x\geq1, \end{cases} $$ a piecewise linear function. Does the following hold for $g(x) = 1/x$? $$ \begin{align} g(f(x)) ...
2
votes
3answers
34 views

what function fulfills these conditions? [duplicate]

So I know that if $f(x) = x^{-1}$, than $f(f(x)) = x$ but $f(x)$ is not necessarily $x$. So now, is there $g(x)$ such that $g(g(x)) \neq g(x) \neq x$ but $g(g(g(x))) = x$? If so what is it, else why ...
0
votes
1answer
47 views

Sherman Morrison Formula for hermitian updates

I have a problem in which, in principle I can apply twice Sherman-Morrison formula but it seems to me that for this case, there should be a simpler solution so my question is "May the process ...
0
votes
2answers
58 views

Inverse of 3-by-3 matrix

Hi, so this question is taken straight from khan academy help exercises, i know how to do it dynamically meaning using the determinant and the adjugate how i was trying to do it using guass bla bla ...
0
votes
0answers
14 views

prove that $X$ is invertible if and only if $Y$ is invertible. if $(-1)^i(1+i)x_i^T=y_i$

$X=[x_1,x_2,...,x_n]$ and $Y =$ $y_1\\y_2\\...\\...\\...\\y_n$ where $x_i$ and $y_i$ are column and row matrices respectively. $X$ and $Y$ are both $n$ x $n$ matrices. if $$(-1)^i(1+i)x_i^T=y_i$$ ...
2
votes
1answer
47 views

Find fundamental matrix of a 2x2 matrix with rank 1

$$ x'(t) = \left[\begin{array}{cccc}0&1\\0&t\end{array}\right]x(t)$$ I am having trouble computing the fundamental matrix. I get: $$ x1(t) = x2(0)*exp(0.5t^2) $$ $$ x2(t) = x2(0)*exp(0.5t^2) ...
0
votes
2answers
32 views

Inverse Sine and cosine

$\arcsin(\cos(x))=1/2$ Find $x$. I got $-1/2$ or $2\pi-1/2$, but I don't know the correct answer. I tried graphing unit circle.
3
votes
1answer
57 views

A square matrix with the diagonal and antidiagonal elements different from zero. Looking for some already known property.

I am interested in the properties of a matrix with elements different from zero only on the main diagonal and antidiagonal, like this: $$ \begin{matrix} a & 0 & 0 & h \\ ...
2
votes
1answer
58 views

If we have a square matrix thats invertible, do its row and column space coincide?

If we have a square matrix thats invertible, do its row and column space coincide? Regarding an nxn invertible matrix: -The row space of the matrix is R^n -The column space of the matrix is R^n ...
1
vote
2answers
32 views

compute the inverse function

Assume $h(x)$ is an invertible function. Let $g(x)=2+8h(4x+1)$. Find the inverse of $g$ in terms of $h^{-1}$ So following the usual steps to get the inverse function, I rearranged to get ...
0
votes
1answer
64 views

Show a matrix is invertible [duplicate]

How to show that $$A=\begin{pmatrix}1233&2344&1324&3456\\ 2342&11233&1432&13256\\234132&32432&1234567&43254\\423412&42354&452356&13245\end{pmatrix}$$ ...
0
votes
4answers
74 views

Calculate the multiplicative inverse modulo a composite number

I want to calculate $ 8^{-1} \bmod 77 $ I can deduce $ 8^{-1} \bmod 77$ to $ 8^{59} \bmod 77 $ using Euler's Theorem. But how to move further now. Should i calculate $ 8^{59} $ and then divide ...
1
vote
1answer
66 views

Summation of $\tan^{-1}$ series

I am given $$S=\sum\limits_{n=1}^{23}\cot^{-1}\left(1+ \sum\limits_{k=1}^n 2k\right)$$ On expanding the sigma series becomes $$S= 23\cot^{-1}(3)+22\cot^{-1}(5) + \cdots + \cot^{-1}(47)$$ And in tan ...
4
votes
2answers
120 views

How to find the inverse cosine without a calculator

How to find the inverse of: $$\cos(c)=\frac{1}{3}$$ In other words, i'm trying to solve for c and without a calculator. If it's hard or not possible, then how would you go about solving inverses in ...
0
votes
7answers
1k views

Proof: The inverse of the inverse matrix is the matrix.

If $A$ is a square matrix such that it is not singular, then $(A^{-1})^{-1} = A$ How can I prove this property? I would appreciate it if somebody can help me.